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Limit of a sequence - Formal definition

Limit of a sequence - Formal definition: Encyclopedia II - Limit of a sequence - Formal definition

Suppose x1, x2, ... is a sequence of elements in a topological space T. We say that L∈T is a limit of this sequence and write if and only if for every neighborhood S of L there is an N such that xn∈S for all n>N. If a sequence has a limit, we say the sequence is convergent, and that the sequence converges to the limit. Otherwise, the sequence is ...

Limit of a sequence: Encyclopedia II - Limit of a sequence - Formal definition

Limit of a sequence - Formal definition

Suppose x₁, x₂, ... is a sequence of elements in a topological space T. We say that L∈T is a limit of this sequence and write

if and only if

for every neighborhood S of L there is an N such that x_n∈S for all n>N.

If a sequence has a limit, we say the sequence is convergent, and that the sequence converges to the limit. Otherwise, the sequence is divergent.

for normed spaces : if ||.|| is the norm, if and only if for every e>0,there is a natural number N so that for every n>N, we have ||x_n-L||<e

for metric spaces : if d is the distance,if and only if for every e>0,there is a natural number N so that for every n>N, we have d(x_n,L)<e

Limit of a sequence - Comments

The definition means that eventually all elements of the sequence get as close as we want to the limit. (This does not imply that there is a limit whenever the elements become as close as we want to all of the following elements, see Cauchy sequence).

Also, a sequence may have several different limits, but a convergent sequence has a unique limit if T is a Hausdorff space, as for example the (extended) real line, the complex plane, their subsets (R, Q, Z...) and Cartesian products (Rⁿ...).